I believe that the TSA security check is not related to a visa - the two are separate. For example I now hold a Green Card although I initially came here on an H1-B (work) visa so I reside here full time. I still had to go through the TSA security check. As pointed out there are 3 rating for which you need the TSA clearance. I suspect you would still need a visa to follow other courses (I doubt you can do it on a the WB/WT program) but would not need clearance from the TSA.
No idea about hour building - probably the best thing is to double check with some flight schools and see what they say.